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calculate-0-t-2-e-t-1-dt-interms-of-3-




Question Number 53783 by maxmathsup by imad last updated on 25/Jan/19
calculate ∫_0 ^∞  (t^2 /(e^t −1))dt interms of ξ(3)
calculate0t2et1dtintermsofξ(3)
Answered by Smail last updated on 26/Jan/19
A∫_0 ^∞ (t^2 /(e^t −1))dt=∫_0 ^∞ ((t^2 e^(−t) )/(1−e^(−t) ))dt  =∫_0 ^∞ t^2 e^(−t) Σ_(n=0) ^∞ e^(−nt) dt  =Σ_(n=0) ^∞ ∫_0 ^∞ t^2 e^(−(n+1)t) dt  by parts  u=t^2 ⇒u′=2t  v′=e^(−(n+1)t) ⇒v=((−1)/(n+1))e^(−(n+1)t)   A=Σ_(n=0) ^∞ (2/(n+1))∫_0 ^∞ te^(−(n+1)t) dt  with  ([t^2 e^(−(n+1)t) ]_0 ^∞ =0)  by parts  A=Σ_(n=0) ^∞ (2/((n+1)^2 ))∫_0 ^∞ e^(−(n+1)t) dt  =Σ_(n=0) ^∞ (2/((n+1)^3 ))[e^(−(n+1)t) ]_0 ^∞ =2Σ_(n=0) ^∞ (1/((n+1)^3 ))  =2Σ_(n=1) ^∞ (1/n^3 )=2ξ(3)
A0t2et1dt=0t2et1etdt=0t2etn=0entdt=n=00t2e(n+1)tdtbypartsu=t2u=2tv=e(n+1)tv=1n+1e(n+1)tA=n=02n+10te(n+1)tdtwith([t2e(n+1)t]0=0)bypartsA=n=02(n+1)20e(n+1)tdt=n=02(n+1)3[e(n+1)t]0=2n=01(n+1)3=2n=11n3=2ξ(3)
Commented by maxmathsup by imad last updated on 26/Jan/19
thank you sir
thankyousir

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